Answer a,b and c Transcribed Image Text: 2.Č Let V be an inner product space and v1, , Vn E V.

Excerpt
v;) O for 1 < i + j < n and (v;, v;) = i for 1 < i < n, then V1, V2, · · · , Vn are linearly independent. ... (f) If S is a subset of V, then (S) = S.

Answer a,b and c Transcribed Image Text: 2.Č
Let V be an inner product space and v1,
, Vn E V. For each of

the following statements,
• write (T) if it is true and give a short proof, or
• write (F) if it is false and give a concrete counterexample.
(a) If every vector v E V is expressed as a linear combination of V1,·… , Vn

in a unique way, then dim V = n.
(b) If U and W are subspaces of V, then F = {u – w : u E U, w E W} is a

subspace of V.
(c) If U and W are subspaces of V such that u + w E U UW for all u E U
and w E W, then UUW = U or U U W = W.
(d) If U OW1 = UOW2 where U, W1, W2 are subspaces of V, then W1 = W2.
(e) If (vi, v;)
O for 1 < i + j < n and (v;, v;) = i for 1 < i < n, then V1, V2, · · · , Vn are linearly independent. ... (f) If S is a subset of V, then (S) = S.

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